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Rational functions are expressions of the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \). These functions can be thought of as similar to fractions, where the numerator and denominator are polynomial expressions.

One key aspect to remember is that the denominator cannot be zero, as it would make the function undefined. This is why identifying the roots of the denominator is essential when determining the domain. You have to exclude those values from the set of possible inputs.

Rational functions can exhibit complex behavior at these points, like vertical asymptotes, so understanding where the function is undefined helps in sketching the general behavior of the graph.

Factoring polynomials is a powerful tool used to simplify expressions and solve equations. To find the domain of a rational function, you often need to factor the polynomial in the denominator to find its roots. These roots are the values of \( x \) that make the denominator equal to zero.

Here's a quick way to factor a polynomial like \( x^3 - 2x^2 - 9x + 18 \).

• Look for patterns or use synthetic division to simplify.
• Once factored, you get expressions like \( (x-2)(x-3)(x+3) \).
• Set each factor to zero to find the roots: \( x = 2, x = 3, \) and \( x = -3 \).

By solving these, you find the values that need to be excluded from the domain. Factoring is essential for dealing with more complex polynomials, as it breaks them down into more manageable pieces.

Interval notation is a compact way to describe a set of numbers, particularly useful when defining domains. After finding the values that make a rational function's denominator zero, use interval notation to express all other numbers in the domain.

For the function given, where \( x = 2, 3, \) and \( -3 \) are excluded, the domain in interval notation is written as:
\((-\infty, -3) \cup (-3, 2) \cup (2, 3) \cup (3, \infty)\).

This notation uses parentheses \(()\) to indicate that the endpoints are not included in the set and the symbol \(\cup\) (union) to combine intervals. It's an efficient way to list all permissible values for \( x \) in a clear, easy-to-read format. Understanding interval notation is crucial for correctly interpreting and expressing mathematical solutions.